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The bounded vector measure associated to a conical measure and pettis differentiability

Part of: Measures, integration, derivative, holomorphy Set functions, measures and integrals with values in abstract spaces

Published online by Cambridge University Press:  09 April 2009

L. Rodriguez-Piazza  and
M. C. Romero-Moreno
L. Rodriguez-Piazza
Affiliation:
Departamento de Análisis Matemático Facultad de Matemáticas Universidad de SevillaAptdo. 1160 Sevilla 41080Spain e-mail: piazza@cica.es, mcromero@cica.es
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Abstract

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Let X be a locally convex space. Kluvánek associated to each X-valued countably additive vector measure a conical measure on X; this can also be done for finitely additive bounded vector measures. We prove that every conical measure u on X, whose associated zonoform Ku is contained in X, is associated to a bounded additive vector measure σ(u) defined on X, and satisfying σ(u)(H) ∈ H, for every finite intersection H of closed half-spaces. When X is a complete weak space, we prove that σ(u) is countably additive. This allows us to recover two results of Kluvánek: for any X, every conical measure u on it with Ku ⊆ X is associated to a countably additive X-valued vector measure; and every conical measure on a complete weak space is localizable. When X is a Banach space, we prove that σ(u) is countably additive if and only if u is the conical measure associated to a Pettis differentiable vector measure.

Keywords

vector measuresconical measuresranges of vector measuresPettis integral

MSC classification

Secondary: 46G10: Vector-valued measures and integration 28B05: Vector-valued set functions, measures and integrals
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 70 , Issue 1 , February 2001 , pp. 10 - 36
Copyright
Copyright © Australian Mathematical Society 2001

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